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A309071
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Complete list of solutions to y^2 = x^3 + 20*x; sequence gives x values.
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1
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OFFSET
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1,2
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COMMENTS
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(x,y) = (0,0) is this solution. Consider x > 0. If x is square, x^2 + 20 is square and we get (x,y) = (4,12). If x is not square, x = i^2*j where j is squarefree. j | x^2 + 20, so j is 2,5 or 10. If j = 2 or 10, there is no such (x,y). If j = 5, (y/5)^2 = i^2*(5*i^4 + 4). So 5*i^4 + 4 = k^2. That is k^2 - 5*i^4 = 4. i^2 is a square Fibonacci number. i^2 = 1 or 144, so x = 5 or 720.
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LINKS
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EXAMPLE
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0^3 + 20* 0 = 0 = 0^2.
4^3 + 20* 4 = 144 = 12^2.
5^3 + 20* 5 = 225 = 15^2.
720^3 + 20*720 = 373262400 = 19320^2.
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PROG
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(PARI) for(k=0, 1e5, if(issquare(k*(k^2+20)), print1(k", ")))
(SageMath) [i[0] for i in EllipticCurve([20, 0]).integral_points()] # Seiichi Manyama, Aug 25 2019
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CROSSREFS
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KEYWORD
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nonn,fini,full
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AUTHOR
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STATUS
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approved
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